18-660: Numerical Methods for Engineering Design and Optimization Xin Li Department of ECE Carnegie Mellon University Pittsburgh, PA 15213
Slide 1
Overview
Introduction Numerical computation Examples and applications
Slide 2
Numerical Computation
Numerical computation: approximation techniques and algorithms to numerically solve mathematical problems
Not all mathematical problems have closed-form solutions
Mathematical examples Solve nonlinear algebraic equations Find minimum of a nonlinear function Determine eigenvalues of a matrix
This course will encompass various aspects of numerical methods and their applications
Linear & nonlinear solver, nonlinear optimization, randomized algorithm, etc. Slide 3
Numerical Computation
Numerical computation is an important tool to solve many practical engineering problems PageRank finds the eigenvector of a 1010-by-1010 matrix – the largest matrix computation in the world
Airlines use optimization algorithms to decide ticket prices, airplane assignments and fuel needs
Slide 4
Numerical Computation
Several additional real-world examples: Car companies run computer simulations of car crashes to identify safety issues
Wall street runs statistical simulation tools to predict stock prices
Slide 5
Brief History of Numerical Computation
Before 1950s
Algorithms Linear interpolation Newton’s method Gaussian elimination
Tools Hand calculation Formula handbook Data table
Carl Friedrich Gauss (1777-1855) German Mathematician
Slide 6
Brief History of Numerical Computation After 1950s
The invention of modern computers motivates the development of a great number of new numerical algorithms 1E+11
1E+10 1E+09
1E+09 1E+08
1E+08
1E+07
1E+07
1E+06 1E+06
CPU Frequency (Hz)
1E+10
1E+05
IPS
2008
2007
2006
2006
2005
2005
2002
2000
1999
1996
1994
1990
1986
1984
1979
1E+05
1974
1E+04
1971
Instruction Per Second
Freq
Slide 7
Numerical Computation Problems
Ordinary and partial differential equations
Regression
Classification
Slide 8
Ordinary Differential Equation (ODE)
Transient analysis for electrical circuit u (t ) dv(t ) = u (t ) − v(t ) dt dv(t ) v(t ) − v(t − ∆t ) ≈ dt ∆t